3.5.1.3ElectricityCore

Resistivity

Why does a long, thin wire have more resistance than a short, thick one made of the same metal? And why does a thermistor’s resistance do the exact $opposite$ of a metal wire’s when it warms up? Both questions come down to a single material property — $resistivity$ — and what’s happening to the charge carriers inside as conditions change.

35 min Video by Science Shorts 3.5.1 Current electricity

Resistivity — A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the resistivity equation $R = \frac{ρ L}{A}$
Explain why resistance depends on a conductor’s length and cross-sectional area
Explain how the resistance of a metal changes with temperature
Explain how the resistance of an NTC thermistor and an LDR change with temperature and light respectively
Describe superconductivity and the significance of the critical temperature

The resistivity equation

A conductor’s resistance depends not just on the material it’s made of, but on its shape: a longer wire has more resistance (charge carriers must travel further, colliding more times), while a wire with a larger cross-sectional area has less resistance (there is more room for current to flow through, like a wider pipe). $Resistivity$ , $ρ$ , is the property that describes only the material itself, independent of its shape, measured in ohm-metres ( $Ω m$ ).

R = \frac{ρ L}{A}

Resistance from resistivity, length and cross-sectional area.

1Use

R = \frac{ρ L}{A}

R = \frac{1.7 \times 1 0 ^{- 8} \times 2.0}{5.0 \times 1 0 ^{- 7}}

R = 0.068 Ω

Answer

R = 0.068 Ω

Tip — Stretching a wire affects resistance twice over — length increases directly, and area decreases at the same time (if volume is conserved), so effects compound rather than cancel.

Temperature and the resistance of a metal

In a metal, the number of free (conduction) electrons is essentially fixed and doesn’t change much with temperature. As temperature rises, the positive ions in the metal’s lattice vibrate more vigorously, so free electrons collide with them more often as they drift through the material. More frequent collisions mean more resistance — this is why a metal’s resistivity, and hence resistance, $increases$ with temperature.

Tip — This same effect is exactly why a filament lamp’s I–V graph curves (from the previous lesson) — as current heats the filament, its resistance rises.

Thermistors and LDRs

An $NTC (negative temperature coefficient) thermistor$ behaves the opposite way to a metal: as its temperature rises, many more charge carriers become available (electrons are freed from bonds within the semiconductor material) — and this large increase in the number of charge carriers outweighs the increased collision rate, so the thermistor’s resistance $decreases$ sharply as temperature rises.

A $light-dependent resistor (LDR)$ works on the same underlying idea but with light instead of heat: more incident light frees more charge carriers within the semiconductor, so an LDR’s resistance $decreases$ as light intensity increases.

Tip — Metal: resistance up with temperature. NTC thermistor: resistance down with temperature. LDR: resistance down with light intensity. Learn this trio as a contrasting set.

Superconductivity

Certain materials, cooled below a very low $critical temperature$ , undergo a dramatic transition: their resistivity drops to exactly $zero$ . Below this critical temperature, a current can flow through a superconducting loop indefinitely with no energy dissipated as heat at all — a genuinely different regime from just "very low resistance".

Practical uses exploit this zero-resistance state to carry very large currents without loss, most notably in the powerful electromagnets used in MRI scanners and particle accelerators. The major practical obstacle is that most known superconductors need cooling to extremely low temperatures (often using liquid helium), which is itself expensive and technically demanding.

Tip — Don’t confuse "very good conductor" with "superconductor" — a superconductor has genuinely zero resistivity below its critical temperature, not just a very small one.

Equation recap

R = \frac{ρ L}{A}

Resistance from resistivity, length and cross-sectional area.

Common mistakes to avoid

Assuming resistivity itself changes if you cut a wire in half.

Resistivity is a property of the material only; cutting the wire changes its resistance (via L and A) but not its resistivity.

Thinking a thermistor’s resistance increases with temperature, like a metal.

An NTC thermistor’s resistance DECREASES with rising temperature — the opposite behaviour to a metal — because far more charge carriers become available.

Describing a superconductor as having "very low" resistance.

Below its critical temperature, a superconductor’s resistivity is exactly zero, not merely small — a qualitatively different state, not just an extreme point on the usual scale.

Key takeaways

Resistance depends on shape (R = ρL/A); resistivity ρ depends only on the material.
A metal’s resistance increases with temperature (more frequent electron–lattice collisions).
An NTC thermistor’s resistance decreases sharply with temperature; an LDR’s resistance decreases with light intensity (more charge carriers freed in both cases).
Below its critical temperature, a superconductor has exactly zero resistivity.

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Resistivity

Why does a long, thin wire have more resistance than a short, thick one made of the same metal? And why does a thermistor’s resistance do the exact

opposite

of a metal wire’s when it warms up? Both questions come down to a single material property —

resistivity

— and what’s happening to the charge carriers inside as conditions change.

35 min Video by Science Shorts 3.5.1 Current electricity